A Convex Decomposition Formula for the Mumford-Shah Functional in Dimension One

TL;DR

This paper introduces a convex lift of Mumford-Shah functionals in one dimension, establishing a coarea formula and demonstrating the equivalence of minimization problems, leading to existence results for calibrations.

Contribution

It presents a new convex decomposition formula for Mumford-Shah functionals in one dimension, connecting geometric measure theory with variational analysis.

Findings

Proved a generalized coarea formula in dimension one.

Established equivalence between Mumford-Shah minimization and its convex lift.

Obtained a weak existence result for calibrations in one dimension.

Abstract

We study the convex lift of Mumford-Shah type functionals in the space of rectifiable currents and we prove a generalized coarea formula in dimension one, for finite linear combinations of SBV graphs. We use this result to prove the equivalence between the minimum problems for the Mumford-Shah functional and the lifted one and, as a consequence, we obtain a weak existence result for calibrations in one dimension.